derivative of 1/(2sqrt(x+7))

解

\frac{d}{d x} (\frac{1}{2 x + 7})

- \frac{1}{4 ( x + 7 ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{1}{2 x + 7})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{2} \frac{d}{d x} (\frac{1}{x + 7})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= \frac{1}{2} \frac{d}{d x} ((x + 7)^{- \frac{1}{2}})

連鎖法則を適用:

- \frac{1}{2 ( x + 7 ) ^{\frac{3}{2}}} \frac{d}{d x} (x + 7)

= - \frac{1}{2 ( x + 7 ) ^{\frac{3}{2}}} \frac{d}{d x} (x + 7)

\frac{d}{d x} (x + 7) = 1

= \frac{1}{2} (- \frac{1}{2 ( x + 7 ) ^{\frac{3}{2}}} \cdot 1)

簡素化

\frac{1}{2} (- \frac{1}{2 ( x + 7 ) ^{\frac{3}{2}}} \cdot 1) : - \frac{1}{4 ( x + 7 ) ^{\frac{3}{2}}}

= - \frac{1}{4 ( x + 7 ) ^{\frac{3}{2}}}